Why did David specify his mule?
Rashi: It is a sign that Shlomo will rule, for a commoner may not ride on the king's horse. Malbim - a commoner may not even use the king's Kelim!
What is "Gichon"?
Rashi, Targum Yonasan: It is the spring "ha'Shilu'ach" (mentioned in Yeshayah 8:6). From here we learn that we anoint kings only by a spring, for a Siman that their kingship continue. Malbim -ha'Shilu'ach flows slowly, for a Siman that his kingship will be serene.
What does "Avdei Adoneichem" refer to?
Radak: It is the Kereisi u'Pleisi (refer to Shmuel II, 8:18:1:1-5 and the notes there). It is as if he said Avadai. Also "v'El Moshe Amar Ale El Hashem" (Shemos 24:1) is like 'Elai"; "v'Es Yiftach v'Es Shmuel" (Shmuel I, 12:11) means 'v'Osi'.